Analytic Launch Collision Avoidance Methodology

ABSTRACT

An analytic launch collision avoidance method includes providing a launch collision avoidance report generated by evaluating launch times for a launch vehicle and candidate satellite crossing times for a plurality of satellites based on a geometric formulation in terms of two time variables.

TECHNICAL FIELD

The invention relates generally to navigating an environment of residentspace objects and, in particular, to a launch collision avoidancemethodology in which safe launch times are determined using a completelyanalytic approach based on a geometric formulation in terms of two timevariables.

BACKGROUND ART

When a satellite is launched, it and the associated launch vehiclestages must pass through an environment of resident space objects.Depending on the final altitude of the new launch, its trajectory couldpotentially pass through the domain of thousands of resident satellites.Because the launch may occur any time during a launch window, it is avery complicated problem to distinguish safe launch times from unsafelaunch times. The launch window span may range from a few minutes to afew hours. The planned launch usually follows a flight profile specifiedin an Earth fixed coordinate frame.

The process of launch collision avoidance conventionally involvescomparing a flight profile with the positions of the resident satellitepopulation for times throughout the launch window. In this way, timeperiods during the launch window are identified for which there isminimal risk of collision with any resident satellite. A selected launchtrajectory should avoid any satellite close approach which creates anunacceptable level of risk, e.g., a probability of collision greaterthan some acceptable threshold.

Various approaches are currently in use to provide launch collisionavoidance, typically, by comparing the launch trajectories of a launchvehicle (for numerous discrete launch times) to trajectories of theresident satellite population. By way of example, in a known approach, alaunch window of some duration is divided into multiple discrete launchtimes spanning the window (either every minute or every second dependingon the resolution required). For each discrete launch time, astraightforward brute force method steps along each satellite orbit todetermine if any satellite “close approaches” occur. The commerciallyavailable software application STK/Conjunction Analysis Tools (STK/CAT),from Analytical Graphics, Inc. of Exton, Pa., performs launch collisionavoidance and includes a close approach analysis feature. See, U.S. Pat.No. 6,102,334.

Unfortunately, conventional launch collision avoidance technologies arecomputationally intensive, as well as inefficient, and the demands uponthem stand to (or may already) exceed their computational capabilitiesor timeline constraints as the satellite catalog and launch frequencygrow.

It would be useful to be able to provide a launch collision avoidancetechnology capable of rapidly and efficiently determining safe launchtimes. It would also be useful to be able to provide a moresophisticated and efficient launch collision avoidance technology thatavoids one or more of the deficiencies of prior solutions, such asreliance upon brute force computational approaches.

SUMMARY OF THE INVENTION

Example embodiments described herein involve an analytic approach to thelaunch screening problem. Through a combination of geometry and dynamicsconsiderations, the determination of launch window closure times isreduced to a relatively small set of discrete time pairs of launch andprediction times. These candidate times are refined with a Newtonminimization using completely analytic partial derivatives. Thismethodology allows treatment of launch collision as a continuousfunction of time in contrast to current methods which can only provideanswers for discrete fixed launch times. The launch collision avoidancemethodology described herein is fast, efficient and accurate with a timesavings approximately two orders of magnitude faster than currentmethods. Moreover, the launch collision avoidance methodology canenhance or augment conventional launch collision avoidance productsmaking them more efficient and applicable to a broader problem set.

In an example embodiment, launch collision avoidance software isconfigured to search for the minimum value of the separation distancebetween the launch vehicle and the satellite, which is a function of thetime of launch and the real world time. The search uses Newton's methodfor a function of two time variables. At the initial estimate, thegradient of the distance function is evaluated using analytical partialderivatives. These computations provide the data needed to computecorrections using Newton's method for both the launch time and the realworld time. These corrections are applied to the initial estimate toprovide an improved estimate for both the launch time and the real worldtime. Then the distance function at the new pair of times is evaluated.The new times are then used to repeat the whole process. The processcontinues until the corrections to both the launch time and the realworld time are insignificant (fractions of a second). At this point, theiteration is declared to have converged resulting in a final estimate ofthe launch time and real world time. That is, they are the pair of timesfor which the launch vehicle and the satellite come closest together.The resulting distance is then assessed to determine whether thesatellite is in danger of a collision.

In an example embodiment, an analytic launch collision avoidance methodincludes providing a launch collision avoidance report generated byevaluating launch times for a launch vehicle and candidate satellitecrossing times for a plurality of satellites based on a geometricformulation in terms of two time variables.

In an example embodiment, an analytic launch collision avoidance methodincludes evaluating launch times for a launch vehicle and candidatesatellite crossing times for a plurality of satellites using completelyanalytic partial derivatives to determine unsafe launch times, andinitiating or canceling a launch event depending upon the unsafe launchtimes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graphical depiction of launch window inertial planes;

FIG. 2 is a graphical depiction of launch plane angles;

FIG. 3 is a graphical depiction of satellite plane intersection angles;

FIG. 4 is a graphical depiction of a launch vulnerability band;

FIG. 5 illustrates a launch collision avoidance methodology inoperation;

FIG. 6 is a process flow illustrating data preprocessing steps in anexample launch collision avoidance methodology;

FIG. 7 is a process flow illustrating main processing steps in anexample launch collision avoidance methodology; and

FIG. 8 illustrates the standard deviation of the launch plane.

DISCLOSURE OF INVENTION

Let the launch window be described by

T _(start)=launch window start time(real world time)

T _(end)=launch window end time(real world time)

where the convention is introduced that an upper case T will be used fortime as measured in the real world (year, day, hour, min, sec) and alower case t will be used for time measured relative to some fixed realworld event time such as element set epoch, launch window start time,etc.

The launch ephemeris is described by a time ordered set of positions andvelocities of the launch vehicle expressed in Earth Centered Fixed (ECF)coordinates. Herein, it is assumed that the launch ephemeris begins onthe surface of the Earth and terminates somewhere along the finaldesired orbit of the satellite. This termination may be the injectionpoint or may be some number of minutes, say 100, beyond the injectionpoint. By extending the ephemeris beyond the injection point, thisassures not only a safe passage to orbit, but at least a safe firstrevolution through the satellite orbit. For satellites launched to ageosynchronous orbit, there may be slightly different ECF ephemerisfiles throughout the launch window because of a requirement forsatellite placement at a specific longitude.

In example embodiments described herein, the launch collision avoidancemethodology is configured (or programmed) to find all clear times withinthe launch window for which the launch trajectory (or trajectories) willnot come closer than a specified threshold, D, to any residentsatellite. It should be further understood that the principles describedherein are also applicable to other space objects such as debris.

FIG. 5 illustrates the launch collision avoidance methodology inoperation. In an operating environment 500, launch collision avoidancesoftware (including launch collision avoidance algorithms) is executedby a computer 502 (including a processor) to generate launch collisionavoidance reports, indications, commands and other inputs, signals orthe like that can be used to make decisions regarding and control launchevents. As illustrated in the left half of the figure, the launchcollision avoidance software has determined that a proposed launch timefor launch vehicle 504 is unsafe because of an unacceptably highprobability of a collision with satellite 506. As also illustrated inthe right half of the figure, the launch collision avoidance softwarehas determined that a slightly later launch time for launch vehicle 504is safe because of an acceptably low probability of a collision withsatellite 506.

Referring to FIGS. 6 and 7, in an example embodiment, a launch collisionavoidance process flow includes data preprocessing 600 and mainprocessing 700.

With respect to the data preprocessing 600, inputs 602 are accessed bythe computer 502 or other processor(s). By way of example, the inputsinclude (but are not limited to): launch window start time, launchwindow end time, risk threshold, distance threshold, file name of launchephemeris, and file name of protected satellites. At 604, the launchephemeris files 606 are read. At 608, rotation is made to the inertialcoordinate system at launch window start time. At 610, the datapreprocessing advances to the step of computing and storing launchephemeris interpolating coefficients 612. Thereafter, at 614, theminimum and maximum ephemeris altitudes are determined. At 616, theplane of the launch ephemeris is determined. At 618, prior to advancingto the main processing, the geometrical parameters of the launch windoware computed.

With respect to the main processing 700, the geometrical parameters ofthe launch window output by the data preprocessing 600 are accessed,along with data from the protected satellite file 704. At 702, theprocessing loops as shown through an analysis of all of the satellitesin the protected satellite file 704. At 706, if it is determined thatthere are data (for additional satellites) that remains to be evaluated,the main processing 700 advances to the perigee apogee filter 708, whichis applied to the satellite data. More specifically, the perigee apogeefilter 708 compares a launch vehicle trajectory to the perigee andapogee of satellites to eliminate satellites from further analysis. If adetermination is made, at 710, that the satellite is not within analtitude rage of concern, then the main processing 700 loops back toanalyze the next satellite in the protected satellite file 704. If,however, the satellite is within an altitude rage of concern, then themain processing 700 advances, at 712, to computing candidate crossingtimes for intersections of the satellite passage windows and the launchvehicle passage windows. At 714 and 716, the main processing 700 loopsthrough the process of examining all candidate times. At 718, the timeof minimum distance between the launch vehicle and the satellite isdetermined. If it is determined, at 720, that the minimum distance isnot closer than a distance threshold, then the main processing 700returns to looping through the examination of candidate times. If theminimum distance is closer (i.e., less) than the distance threshold,then the processing advances to 722 where the time span for which thesatellite and launch vehicle are within the distance threshold isdetermined. At 724, the collision probability is computed inconventional fashion. See, e.g., Chan, K., “Improved AnalyticalExpressions for Computing Spacecraft Collision Probabilities,” AAS PaperNo. 03-184, AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico,9-13 Feb. 2003, which is incorporated herein by reference in itsentirety. At 726, if it is determined that the probability of collisionis higher than a risk threshold, at 728, the window risk times 730 arestored. If it is determined that the probability of collision is nothigher than the risk threshold, and after the window risk times arestored, the main processing 700 returns again to looping through theexamination of candidate times. At 732, unsafe launch times aredetermined and used to generate a launch collision avoidance report 734,which includes information such as: closure time, offending satellite,and risk level.

In the following sections, details of an example embodiment of the datapreprocessing 600 and main processing 700 are discussed.

Launch Ephemeris Characterization

Given an ECF launch ephemeris, data preprocessing 600 begins byperforming some preprocessing of the data. This preprocessing is validfor all resident satellites so it need be done only once. Let

{right arrow over (r)}_(ECF)=position in ECF coordinate system

{right arrow over (v)}_(ECF)=velocity in ECF coordinate system

be a point in the ephemeris file at time t relative to the start of theephemeris file. The launch trajectory is provided in Earth fixedcoordinates. It can be transformed into inertial coordinates using arotation matrix so that

{right arrow over (r)}_(L)=Q{right arrow over (r)}_(ECF)

{right arrow over (v)} _(L) =Q{right arrow over (v)} _(ECF) +{dot over(Q)}{right arrow over (r)} _(ECF)

where the subscript L denotes the launch vehicle in an inertialcoordinate system.

$Q = \begin{pmatrix}{\cos \; \theta} & {{- \sin}\; \theta} & 0 \\{\sin \; \theta} & {\cos \; \theta} & 0 \\0 & 0 & 1\end{pmatrix}$ $\overset{.}{Q} = {\overset{.}{\theta}\begin{pmatrix}{{- \sin}\; \theta} & {{- \cos}\; \theta} & 0 \\{\cos \; \theta} & {{- \sin}\; \theta} & 0 \\0 & 0 & 0\end{pmatrix}}$

with

θ=Greenwich hour angle at time T _(start) +t

{dot over (θ)}=rotation rate of the Earth

Next, cubic spline interpolating coefficients for the entire ephemerisfile (or files) are determined. These interpolating coefficients aredetermined only once.

The data preprocessing 600 next examines every point in the ephemerisfile and determines the largest and smallest radial distance achievedanywhere in the ephemeris. Let

r_(High)=largest radial distance in ephemeris

r_(Low)=smallest radial distance in ephemeris

The average plane of the ephemeris at the beginning of the launch windowis then determined. Let

i_(Av)=average inclination of ephemeris

Ω_(Av)=average right ascension of ephemeris

σ_(Plane)=standard deviation of plane of ephemeris

Finally, the data is supplemented with interpolating coefficients forthe true argument of latitude and radial distance of the ephemeris.

Launch Wedge Characterization

As discussed above, the ephemeris foiins an approximate plane at thebeginning of the launch window. During the course of the launch window,this plane rotates with the Earth to a final location at the end of thelaunch window. The beginning and ending planes can be characterized bythe orientation angles

Ω_(start)=Ω_(Av)

i_(start)=i_(Av)

Ω_(end)=Ω_(start)+{dot over (θ)}(T _(end) −T _(start))

i_(end)=i_(Av)

FIG. 1 illustrates what these two planes might look like for a launchwindow of a few hours in length. These two planes form two wedges muchlike two sections of an orange. Let

{right arrow over (w)}_(start)=unit vector normal to the start inertialplane

{right arrow over (w)}_(end)=unit vector normal to the end inertialplane

The line of intersection of the two planes is given by the unit vector

$\overset{\rightarrow}{k} = \frac{{\overset{\rightarrow}{w}}_{start} \times {\overset{\rightarrow}{w}}_{end}}{{{\overset{\rightarrow}{w}}_{start} \times {\overset{\rightarrow}{w}}_{end}}}$

Because these wedges are formed by rotating the launch plane with theEarth, the left hand plane is the start time plane and the right handplane is the end time plane. Therefore, the {right arrow over (k)} unitvector will point to the ascending side of the launch trajectories. Thelaunch trajectories move along the planes until they reach theintersection point where the planes will cross. These two wedges shallbe referred to as the ascending wedge and the descending wedge. Let

Δ_(start)=angle from equator to {right arrow over (k)} measured in startplane

Δ_(end)=angle from equator to {right arrow over (k)} measured in endplane

These two angles will be in the range [0,π] by construction. FIG. 2illustrates these angles. An orbiting satellite is vulnerable tocollision with the launch vehicle only during times in which it iswithin these wedges.

Up to this point all computations have been independent of the satelliteorbit to be avoided. Thus, they need be done only once for the entirelaunch screening.

Altitude Filter

The following is a discussion of a particular satellite being assessedfor launch collision risk for the entire launch window. Launch windowsare typically tens of minutes in length for near Earth satellites and afew hundred minutes in length for geosynchronous satellites. During suchshort time periods, a resident satellite orbit does not undergosignificant perturbations. However, the satellite element set epoch maybe a significant time away from the launch window time. Hence, thesatellite is predicted to the midpoint time of the launch window. Thisprovides updated mean orbital elements which can be assumed to notchange significantly over a time interval of half the launch windowspan.

If the perigee of the satellite orbit is higher than the highestaltitude of the launch trajectory, then the satellite does not need tobe considered. That is, if

a(1−e)>r _(High) +D

then the satellite can be filtered from further consideration.

Likewise, if the apogee of the satellite is lower than the lowestaltitude of the launch trajectory, then the satellite does not need tobe considered. That is, if

a(1+e)+D<r _(Low)

then the satellite can be filtered from further consideration.

If an orbit has not been filtered by either of the altitude filters,then it is assumed that it may potentially intersect or come close tothe launch trajectory for some launch time.

Satellite Time Windows

A satellite is only vulnerable to collision with the launch vehicleduring times in which the satellite is passing through the launch wedge.Such time periods shall be referred to as satellite time windows.

The following notation is now introduced for the mean orbital elementsof the satellite predicted to the midpoint of the launch window.

n=mean motion

e=mean eccentricity

i=mean inclination

Ω=mean right ascension of ascending node

ω=mean argument of perigee

M=mean anomaly

The satellite epoch time, T₀ (real world time), is also introduced,where the subscript 0 convention indicates the value of a variable atthe element set epoch time.

In order to determine the times in which the satellite is passingthrough the launch wedge, continue the development of a geometric modelof the satellite-launch wedge relationship. The line of intersection ofthe satellite plane and the launch start plane is given by the vector

${\overset{\rightarrow}{k}}_{start} = \frac{{\overset{\rightarrow}{w}}_{Sat} \times {\overset{\rightarrow}{w}}_{start}}{{{\overset{\rightarrow}{w}}_{Sat} \times {\overset{\rightarrow}{w}}_{start}}}$

where

{right arrow over (w)}_(Sat)=unit normal to satellite plane at midpointof launch window

Note that the previous equation is not defined when the satellite planeand the launch plane coincide. This is called the coplanar case and mustbe handled separately. It will be discussed in a later section.

Let

u_(Sat) _(—) _(start)=angle from equator to {right arrow over(k)}_(start) measured in satellite plane

u_(launch) _(—) _(start)=angle from equator to {right arrow over(k)}_(start) measured in launch start plane

Then u_(Sat) _(—) _(start) is the satellite true argument of latitude atwhich the satellite enters the vulnerability wedge. FIG. 3 illustratesthese angles.

Let

u₁=u_(Sat) _(—) _(start)

Similarly, the line of intersection of the satellite plane and thelaunch end plane is given by the vector

${\overset{\rightarrow}{k}}_{end} = \frac{{\overset{\rightarrow}{w}}_{Sat} \times {\overset{\rightarrow}{w}}_{end}}{{{\overset{\rightarrow}{w}}_{Sat} \times {\overset{\rightarrow}{w}}_{end}}}$

Let

u_(Sat) _(—) _(end)=angle from equator to {right arrow over (k)}_(end)measured in satellite plane

u_(launch) _(—) _(end)=angle from equator to {right arrow over(k)}_(end) measured in launch end plane

The angle from the equator to the line of intersection with the endplane will be the true argument of latitude of the satellite when itexits the vulnerability band. Let

u₂=u_(Sat) _(—) _(end)

Once the entry and exit true arguments of latitude are obtained, theother entry and exit can be computed by adding π.

In this way, two intervals are formed

[u₁,u₂] [u₃,u₄]

which are the satellite true argument of latitude intervals when thesatellite is passing through the launch vulnerability wedges.

The launch wedge was formed by approximating the plane followed by thelaunch trajectory. A standard deviation of the launch planeapproximation is also computed. This statistic can be used to extend thesatellite angle windows to account for the launch wedge approximationwith a 3 sigma level of confidence.

The launch trajectory will not usually be confined to an exact plane.However, it can be approximated by an average plane. The actualtrajectory will vary about this average plane. This variation isquantified by the standard deviation of the launch plane. Referring toFIG. 8, let

σ_(plane)=standard deviation of launch plane

I_(R)=relative inclination between launch and satellite planes

The dotted line represents the one-sigma noise in the launch orbitalplane. Then

${\Delta\theta} = \frac{\sigma_{plane}}{\sin \; I_{R}}$

is the amount of in-track expansion required for the satellite to reachthe one-sigma plane noise.

This angle increment should be subtracted from the satellite entry angleand added to the satellite exit angle to produce an expanded satellitewindow that allows for the noise in the orbital plane. A three sigmavalue can be used to assure that the window is sufficiently expanded.

Thus, it has been demonstrated that

${\Delta \; u} = {3\frac{\sigma_{Plane}}{\sin \; I_{R}}}$

so the expanded satellite angle windows become

u ₁ *=u ₁ −Δu

u ₂ *=u ₂ +Δu

u ₃ *=u ₃ −Δu

u ₄ *=u ₄ +Δu

The corresponding true anomalies, f_(j), are found using

f _(j) =u _(j)*−ω

Then find the corresponding eccentric anomalies, E_(j), using

$E_{j} = {2{\tan^{- 1}\left( {\sqrt{\frac{1 - }{1 + }}{\tan \left( \frac{f_{j}}{2} \right)}} \right)}}$

The corresponding mean anomalies, M_(j), are given by

M _(j) =E _(j) −e sin E _(j)

Then the times of flight from perigee passage to the desired latitudewill be

$t_{j} = \frac{M_{j}}{n}$

At this point, there are two time intervals

[t₁,t₂] [t₃,t₄]

during which the satellite is passing through the launch wedges.

By adding multiples of the satellite period, a series of time windowsduring which the satellite is in the launch wedge can be generated. Butthe in-track motion is affected by the geopotential through J₂ and theatmospheric drag through {dot over (n)}. So the period of the satellitecontinually changes due to atmospheric drag. However, being onlyinterested in the time that it takes for one complete revolution, theperiod of the (i+1)th revolution in terms of the period of the ithrevolution can be approximated as follows.

$P_{i + 1} = \frac{2\pi}{n_{0} + {\overset{.}{M}}_{0} + {{\overset{.}{n}}_{0}P_{i}}}$

where

${\overset{.}{M}}_{0} = {{- \frac{3}{4}}\frac{J_{2}R^{2}{n_{0}\left( {1 - {3\cos^{2}i_{0}}} \right)}}{{a_{0}^{2}\left( {1 - e_{0}^{2}} \right)}^{3/2}}}$

Then a series of latitude band passage time windows (relative time) isgenerated

$t_{1i} = {t_{perigee} + t_{1}^{*} + {\sum\limits_{j = 2}^{i}\; P_{j - 1}}}$$t_{2i} = {t_{perigee} + t_{2}^{*} + {\sum\limits_{j = 2}^{i}\; P_{j - 1}}}$$t_{3i} = {t_{perigee} + t_{3}^{*} + {\sum\limits_{j = 2}^{i}\; P_{j - 1}}}$$t_{4i} = {t_{perigee} + t_{4}^{*} + {\sum\limits_{j = 2}^{i}\; P_{j - 1}}}$

where t_(perigee)=time relative to epoch of the most recent perigeepassage prior to T_(start).

Thus, the time windows relative to epoch during which the satellite isflying through the launch vulnerability band are

[t_(i1),t_(i2)] [t_(i3),t_(i4)]

The intersection of the satellite orbit with the launch wedge creates astrip across the wedge, which shall be referred to as the launchvulnerability band. This is illustrated in FIG. 4.

The band begins when the launch vehicle reaches u_(launch) _(—) _(start)and ends when the launch vehicle reaches u_(launch) _(—) _(end). Becauseof the noise in the plane approximation, the entry and exit launchangles are modified as follows.

u _(launch) _(—) _(start) *=u _(launch) _(—) _(start) −Δu

u _(launch) _(—) _(end) *=u _(launch) _(—) _(end) −Δu

The earliest time that the launch vehicle could be at u_(launch) _(—)_(start)* will be

T _(entry) =T _(start)+time of flight from launch to u _(launch) _(—)_(start)*

The latest time that the launch vehicle could be at u_(launch) _(—)_(end)* will be

T _(exit) =T _(end)+time of flight from launch to u_(launch) _(—)_(end)*

This results in a time window for the launch vehicle[T_(entry),T_(exit)] during which the launch vehicle could be in thelaunch vulnerability band.

Because the launch trajectory may extend for one complete revolution ormore, there could be other times during which the launch vehicle couldbe in the launch vulnerability band on the other side of the orbit. Thatband is located at

u_(launch) _(—) _(start)*+π u_(launch) _(—) _(end)*+π

This will produce a corresponding time window. In a similar way, aseries of launch vulnerability time windows can be generated until theend of the launch ephemeris span is reached.

Window Altitudes

For the time windows in which the satellite is passing through thelaunch wedge

[t₁,t₂] [t₃,t₄]

and the time windows during which the launch vehicle is crossing thepath of the satellite

[T_(entry),T_(exit)],

the altitudes of the satellite and the launch vehicle as they traversethese windows can also be considered.

Let

R_(L-max)=maximum altitude of launch vehicle in crossing window

R_(L-min)=minimum altitude of launch vehicle in crossing window

r_(S-max)=maximum altitude of satellite in crossing window

r_(S-min)=minimum altitude of satellite in crossing window

Then if

R _(L-max) +D<r _(S-min)

or

R _(L-min) >r _(S-max) +D

the passage windows need not be considered.

Time Candidates

The main processing 700 now looks for intersections of the satellitepassage windows and the launch vehicle passage windows. For anyintervals that intersect, a candidate time is created at the midpoint ofthe intersection.

Let t_(sat) be a candidate time relative to the satellite epoch. This isused as a starting point for an iterative search for the time of closestapproach of the launch vehicle to the satellite.

The first estimate for the launch time, in this example, assumes thatthe minimum occurs at the midpoint of the launch ephemeris span. Thus,the first estimate is

t _(launch) =t _(sat) −t _(TOF)

where

$t_{TOF} = \frac{T_{entry} - T_{start} + T_{exit} - T_{end}}{2}$

After the iteration has converged, a solution pair (t_(S),t_(L)) isyielded such that the distance between the launch vehicle and satelliteis minimized.

Distance Minimization

A function that gives the square of the distance between the launchvehicle and the satellite is now defined. Let where

D(t _(S) ,t _(L))=({right arrow over (r)} _(S) −{right arrow over (r)}_(L))·({right arrow over (r)} _(S) −{right arrow over (r)} _(L))

where

{right arrow over (r)} _(S)(t _(S))=position vector of satellite

{right arrow over (r)} _(L)(t _(s))=position vector of launch vehicle

where the function D depends on both the satellite prediction time t_(S)and the launch time t_(L). Because t_(S) and t_(L) are independentvariables, the distance function will reach a minimum or maximum when

$\frac{\partial D}{\partial t_{S}} = 0$$\frac{\partial D}{\partial t_{L}} = 0$

The partial derivative constraint equations are solved simultaneouslyfor the satellite prediction time and the launch time. They can bereadily solved with an iterative technique starting from the very goodfirst estimate from the previous discussion. This iteration produces thelaunch time and the satellite prediction time when the satellite andlaunch vehicle come closest. This time pair (t_(sat),t_(Launch))provides a point of departure for computing other metrics such asprobability of collision.

Analytic Partial Derivatives

As discussed above, the partial derivative constraint equations are tobe solved simultaneously for the satellite prediction time and thelaunch time. By way of example, they can be solved using Newton'siterative method, given the very good first estimate (discussed supra).In particular, Newton's method gives

t _(S(i+1)) =t _(S(i)) +h

t _(L(i+1)) =t _(L(i)) +k

where

$h = \frac{{F\frac{\partial G}{\partial t_{L}}} - {G\frac{\partial F}{\partial t_{L}}}}{{\frac{\partial F}{\partial t_{L}}\frac{\partial G}{\partial t_{S}}} - {\frac{\partial F}{\partial t_{S}}\frac{\partial G}{\partial t_{L}}}}$$k = \frac{{G\frac{\partial F}{\partial t_{S}}} - {F\frac{\partial G}{\partial t_{S}}}}{{\frac{\partial F}{\partial t_{L}}\frac{\partial G}{\partial t_{S}}} - {\frac{\partial F}{\partial t_{S}}\frac{\partial G}{\partial t_{L}}}}$$F = \frac{\partial D}{\partial t_{S}}$$G = \frac{\partial D}{\partial t_{L}}$

This iteration will give the launch time and the satellite predictiontime when the satellite and launch vehicle come closest. This time pair(t_(S),t_(L)) provides a point of departure for computing other metricssuch as probability of collision.

Now the partial derivatives are computed. First, consider the partialderivative with respect to t_(S) for which it is assumed that t_(L) isfixed.

$F = {\frac{\partial D}{\partial t_{S}} = {{2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}} = 0}}$

Now the partial derivative with respect to t_(L) is considered. Becausethe satellite position does not depend on the launch time,

$G = {\frac{\partial D}{\partial t_{L}} = {{2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {\frac{\partial{\overset{\rightarrow}{r}}_{S}}{\partial t_{L}} - \frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}}} \right)}} = {{- 2}{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( \frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} \right)}}}}$

Now,

$\frac{\partial{\overset{->}{r}}_{L}}{\partial t_{L}}$

is considered. The launch trajectory is provided in Earth fixedcoordinates. The inertial position of the launch vehicle is given by

{right arrow over (r)}_(L)=M{right arrow over (W)}

where

$M = \begin{pmatrix}{\cos \; \Omega} & {{- \sin}\; \Omega \; \cos \; i} & {\sin \; \Omega \; \sin \; i} \\{\sin \; \Omega} & {\cos \; \Omega \; \cos \; i} & {{- \cos}\; \Omega \; \sin \; i} \\0 & {\sin \; i} & {\cos \; i}\end{pmatrix}$ $\overset{\rightarrow}{W} = \begin{pmatrix}{r\; \sin \; u} \\{r\; \cos \; u} \\0\end{pmatrix}$

The partial derivative will be

$\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} = {{M\frac{\partial\overset{\rightarrow}{W}}{\partial t_{L}}} + {\frac{\partial M}{\partial t_{L}}\overset{\rightarrow}{W}}}$

The time of flight of the launch vehicle is defined as follows

t _(TOF) =t _(S) −t _(L)

A change in the launch time will cause an opposite change in the time offlight. This effect must also be included in the partial derivative.Note that the change in sign is due to the effect being in the oppositedirection.

$\frac{\partial\overset{\rightarrow}{W}}{\partial t_{L}} = {- \frac{\partial\overset{\rightarrow}{W}}{\partial t_{TOF}}}$

Thus,

$\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} = {{- {\overset{\rightarrow}{v}}_{L}} + {\frac{\partial M}{\partial t_{L}}\overset{\rightarrow}{W}}}$

A change in the launch time will affect the plane orientation matrix M.Accordingly, the partial derivative of the matrix with respect to launchtime is

$\frac{\partial M}{\partial t_{L}} = {M_{L} = {\overset{.}{\theta}\begin{pmatrix}{{- \sin}\; \Omega} & {{- \cos}\; \Omega \; \cos \; i} & {\cos \; \Omega \; \sin \; i} \\{\cos \; \Omega} & {{- \sin}\; \Omega \; \cos \; i} & {\sin \; \Omega \; \sin \; i} \\0 & 0 & 0\end{pmatrix}}}$

where {dot over (θ)} is the rotation rate of the Earth.

Thus, the total partial derivative is

$\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} = {{- {\overset{\rightarrow}{v}}_{L}} + {M_{L}\overset{\rightarrow}{W}}}$

Equivalently,

$\mspace{20mu} {\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} = {{{- {\overset{\rightarrow}{v}}_{L}} + {M_{L}M^{T}M\overset{\rightarrow}{W}}} = {{- {\overset{\rightarrow}{v}}_{L}} + {M_{L}M^{T}{\overset{\rightarrow}{r}}_{L}}}}}$${M_{L}M^{T}} = {{\overset{.}{\theta}\begin{pmatrix}{{- \sin}\; \Omega} & {{- \cos}\; \Omega \; \cos \; i} & {\cos \; \Omega \; \sin \; i} \\{\cos \; \Omega} & {{- \sin}\; \Omega \; \cos \; i} & {\sin \; \Omega \; \sin \; i} \\0 & 0 & 0\end{pmatrix}}\begin{pmatrix}{\cos \; \Omega} & {\sin \; \Omega} & 0 \\{{- \sin}\; \Omega \; \cos \; i} & {\cos \; \Omega \; \cos \; i} & {\sin \; i} \\{\sin \; \Omega \; \sin \; i} & {{- \cos}\; \Omega \; \sin \; i} & {\cos \; i}\end{pmatrix}}$$\mspace{20mu} {{M_{L}M^{T}} = {\overset{.}{\theta}\begin{pmatrix}0 & {- 1} & 0 \\1 & 0 & 0 \\0 & 0 & 0\end{pmatrix}}}$$\mspace{20mu} {N = {{M_{L}M^{T}} = {\overset{.}{\theta}\begin{pmatrix}0 & {- 1} & 0 \\1 & 0 & 0 \\0 & 0 & 0\end{pmatrix}}}}$   So$\mspace{20mu} {\frac{\partial D}{\partial t_{L}} = {{- 2}{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} = {N{\overset{\rightarrow}{r}}_{L}}} \right)}}}$  Then, finally$\mspace{20mu} {F = {\frac{\partial D}{\partial t_{S}} = {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}}}}$$\mspace{20mu} {G = {\frac{\partial D}{\partial t_{L}} = {{- 2}{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}}}}$  Now$\mspace{20mu} {\frac{\partial F}{\partial t_{S}} = {{2{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}} + {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{a}}_{S} - {\overset{\rightarrow}{a}}_{L}} \right)}}}}$$\mspace{20mu} {\frac{\partial F}{\partial t_{L}} = {{{- 2}{\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \frac{\partial{\overset{\rightarrow}{v}}_{L}}{\partial t_{L}}}}}}$

As the partial of the position vector has already been computed, thepartial of the velocity vector is now considered.

{right arrow over (v)}=M{right arrow over (U)}+M{right arrow over (V)}

where

$\mspace{20mu} {\overset{\rightarrow}{U} = \begin{pmatrix}{\overset{.}{r\;}\sin \; u} \\{\overset{.}{r}\; \cos \; u} \\0\end{pmatrix}}$$\mspace{20mu} {\overset{\rightarrow}{V} = \begin{pmatrix}{r\overset{.}{f}\; \cos \; u} \\{{- r}\overset{.}{f}\sin \; u} \\0\end{pmatrix}}$   Then$\mspace{20mu} {\frac{\partial\overset{\rightarrow}{v}}{\partial t_{L}} = {{M\frac{\partial\overset{\rightarrow}{U}}{\partial t_{L}}} + {M\frac{\partial\overset{\rightarrow}{V}}{\partial t_{L}}} + {\frac{\partial M}{\partial t_{L}}\overset{\rightarrow}{U}} + {\frac{\partial M}{\partial t_{L}}\overset{\rightarrow}{V}}}}$  Rearranging  gives$\mspace{20mu} {\frac{\partial\overset{\rightarrow}{v}}{\partial t_{L}} = {{M\left( {\frac{\partial\overset{\rightarrow}{U}}{\partial t_{L}} + \frac{\partial\overset{\rightarrow}{V}}{\partial t_{L}}} \right)} + {\frac{\partial M}{\partial t_{L}}\overset{\rightarrow}{U}} + {\frac{\partial M}{\partial t_{L}}\overset{\rightarrow}{V}}}}$$\mspace{20mu} {\frac{\partial\overset{\rightarrow}{v}}{\partial t_{L}} = {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}}}$  Then$\mspace{20mu} {\frac{\partial F}{\partial t_{L}} = {{{- 2}{\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \frac{\partial{\overset{\rightarrow}{v}}_{L}}{\partial t_{L}}}}}}$$\frac{\partial F}{\partial t_{L}} = {{{- 2}{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}} \right)}}}$$\mspace{20mu} {G = {\frac{\partial D}{\partial t_{L}} = {{- 2}{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}}}}$$\mspace{20mu} {\frac{\partial G}{\partial t_{S}} = {{{- 2}{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {N{\overset{\rightarrow}{v}}_{L}}} \right)}}}}$$\mspace{20mu} {\frac{\partial G}{\partial t_{L}} = {{2{\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}} \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- \frac{\partial{\overset{\rightarrow}{v}}_{L}}{\partial t_{L}}} + {N\frac{\partial{\overset{\rightarrow}{r}}_{L}}{\partial t_{L}}}} \right)}}}}$$\frac{\partial G}{\partial t_{L}} = {{2{\left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- \left( {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}} \right)} + {N\left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} \right)}}}$$\frac{\partial G}{\partial t_{L}} = {{2{\left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} + {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}} + {N{\overset{\rightarrow}{v}}_{L}} - {{NN}{\overset{\rightarrow}{r}}_{L}}} \right)}}}$  In  summary$\mspace{20mu} {F = {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}}}$$\mspace{20mu} {G = {{- 2}{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}}}$$\mspace{20mu} {\frac{\partial F}{\partial t_{S}} = {{2{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}} + {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}} \right)}}}}$$\mspace{20mu} {\frac{\partial G}{\partial t_{S}} = {{{- 2}{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {N{\overset{\rightarrow}{v}}_{L}}} \right)}}}}$$\frac{\partial G}{\partial t_{L}} = {{2{\left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} + {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}} + {N{\overset{\rightarrow}{v}}_{L}} - {{NN}{\overset{\rightarrow}{r}}_{L}}} \right)}}}$  where $\mspace{20mu} {M_{L} = {\overset{.}{\theta}\begin{pmatrix}{{- \sin}\; \Omega} & {{- \cos}\; \Omega \; \cos \; i} & {\cos \; \Omega \; \sin \; i} \\{\cos \; \Omega} & {{- \sin}\; \Omega \; \cos \; i} & {\sin \; \Omega \; \sin \; i} \\0 & 0 & 0\end{pmatrix}}}$$\mspace{20mu} {N = {\overset{.}{\theta}\begin{pmatrix}0 & {- 1} & 0 \\1 & 0 & 0 \\0 & 0 & 0\end{pmatrix}}}$   Because$\mspace{20mu} {F = \frac{\partial D}{\partial t_{S}}}$$\mspace{20mu} {G = \frac{\partial D}{\partial t_{L}}}$  then  it  follows  that$\mspace{20mu} {\frac{\partial F}{\partial t_{L}} = {\frac{\partial^{2}D}{{\partial t_{L}}{\partial t_{S}}} = \frac{\partial G}{\partial t_{S}}}}$  In  comparing$\frac{\partial F}{\partial t_{L}} = {{{- 2}{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}} \right)}}}$  and$\mspace{20mu} {\frac{\partial G}{\partial t_{S}} = {{{- 2}{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} = {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {N{\overset{\rightarrow}{v}}_{L}}} \right)}}}}$  it  is  necessary  to  show  that$\mspace{20mu} {{{M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}} = {N{\overset{\rightarrow}{v}}_{L}}}$  Now$\mspace{20mu} {\overset{\rightarrow}{v} = {{M\overset{\rightarrow}{U}} + {M\overset{\rightarrow}{V}}}}$  and   N = M_(L)M^(T)   so$\mspace{20mu} {{N\overset{\rightarrow}{v}} = {{M_{L}M^{T}M\overset{\rightarrow}{U}} + {M_{L}M^{T}M\overset{\rightarrow}{V}}}}$$\mspace{20mu} {{N\overset{\rightarrow}{v}} = {{M_{L}\overset{\rightarrow}{U}} + {M_{L}\overset{\rightarrow}{V}}}}$

Thus, the mixed second partials agree as they should.

Hence, the results are summarized as follows

F=2({right arrow over (r)} _(S) −{right arrow over (r)} _(L))·({rightarrow over (v)} _(S) −{right arrow over (v)} _(L))

G=−2({right arrow over (r)} _(S) −{right arrow over (r)} _(L))·(−{rightarrow over (v)} _(L) +N{right arrow over (r)} _(L))

$\mspace{20mu} {\frac{\partial F}{\partial t_{S}} = {{2{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right)}} + {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{\overset{\rightarrow}{a}}_{S} - {\overset{\rightarrow}{a}}_{L}} \right)}}}}$$\mspace{20mu} {\frac{\partial F}{\partial t_{L}} = {\frac{\partial G}{\partial t_{S}} - {2{\left( {{\overset{\rightarrow}{v}}_{S} - {\overset{\rightarrow}{v}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} - {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {N{\overset{\rightarrow}{v}}_{L}}} \right)}}}}$$\frac{\partial G}{\partial t_{L}} = {{2{\left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right) \cdot \left( {{- {\overset{\rightarrow}{v}}_{L}} + {N{\overset{\rightarrow}{r}}_{L}}} \right)}} + {2{\left( {{\overset{\rightarrow}{r}}_{S} - {\overset{\rightarrow}{r}}_{L}} \right) \cdot \left( {{- {\overset{\rightarrow}{a}}_{L}} + {2N{\overset{\rightarrow}{v}}_{L}} - {{NN}{\overset{\rightarrow}{r}}_{L}}} \right)}}}$  Now$\mspace{20mu} {{NN} = {{{{\overset{.}{\theta}}^{2}\begin{pmatrix}0 & {- 1} & 0 \\1 & 0 & 0 \\0 & 0 & 0\end{pmatrix}}\begin{pmatrix}0 & {- 1} & 0 \\1 & 0 & 0 \\0 & 0 & 0\end{pmatrix}} = {- {{\overset{.}{\theta}}^{2}\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 0\end{pmatrix}}}}}$

Coplanar Case

Recall that the first step in determining the satellite crossing timesof the launch wedge utilized the following equation.

${\overset{\rightarrow}{k}}_{start} = \frac{{\overset{\rightarrow}{w}}_{Sat} \times {\overset{\rightarrow}{w}}_{start}}{{{\overset{\rightarrow}{w}}_{Sat} \times {\overset{\rightarrow}{w}}_{start}}}$

If the satellite plane and the launch plane coincide, then the equationis not defined. In fact, because the planes coincide, the satellite isalways in the launch wedge. Launch trajectories exist for all timeswithin the launch wedge. So even if the satellite plane does notcoincide with the launch plane at the start of the launch wedge, itmight correspond with the launch plane at a later time in the launchwindow. Thus, a different methodology is needed for any satellites whoseorbital planes are coplanar with the launch plane at any time during thelaunch window. Practically speaking, any satellites whose orbital planesare close to the launch plane at any time during the launch window arealso treated separately.

In an example embodiment, the following test is employed. If

{right arrow over (w)} _(Sat) ·{right arrow over (w)} _(start)>0.95

or

{right arrow over (w)} _(Sat) ·{right arrow over (w)} _(end)>0.95

then the satellite is considered coplanar.

In a launch collision avoidance methodology for the coplanar case,discrete launch times (e.g., at a resolution of one second) areconsidered. Let

D(t _(S))=({right arrow over (r)} _(S) −{right arrow over (r)}_(L))·({right arrow over (r)} _(S) −{right arrow over (r)} _(L))

Then

{dot over (D)}(t _(S))=2({right arrow over (r)} _(S) −{right arrow over(r)} _(L))·({right arrow over (v)} _(S) −{right arrow over (v)} _(L))

For a given discrete launch time, the coplanar approach steps along theorbit in time searching for a change in sign of {dot over (D)} fromnegative to positive. Such a change indicates the distance function haspassed through a minimum and is recorded as a time candidate for furtherinvestigation. The step size can be large (e.g., 20% of satelliteperiod) because the function is smooth. In an example embodiment, onefifth of the smaller of the satellite and launch vehicle orbital periodsis used. A root-finding algorithm, such as Brent's method, can be usedto refine the candidate and determine the time at which the satelliteand launch vehicle come closest for a fixed discrete launch time. Thisresult will be within one second of the true minimum because, in thisexample, fixed launch times at one second spacing were considered. Withthis starting time pair (t_(S),t_(L)), the two variable minimizationtechnique previously described can now be used to find the exact twovariable minimum.

Although the present invention has been described in terms of theexample embodiments above, numerous modifications and/or additions tothe above-described embodiments would be readily apparent to one skilledin the art. It is intended that the scope of the present inventionextend to all such modifications and/or additions.

1. An analytic launch collision avoidance method, comprising the stepof: providing a launch collision avoidance report generated byevaluating launch times for a launch vehicle and candidate satellitecrossing times for a plurality of satellites based on a geometricformulation in terms of two time variables.
 2. The analytic launchcollision avoidance method of claim 1, further comprising the step of:initiating or canceling a launch event depending upon the launchcollision avoidance report.
 3. The analytic launch collision avoidancemethod of claim 1, wherein evaluating the launch times and the candidatesatellite crossing times includes refining a pair of times for which thelaunch vehicle and a satellite come closest together.
 4. The analyticlaunch collision avoidance method of claim 3, wherein the pair of timesis a launch time for the launch vehicle and a real world time when thesatellite and the launch vehicle come closest together.
 5. The analyticlaunch collision avoidance method of claim 3, wherein generating thepair of times includes computing corrections to the pair of times usingan iterative technique.
 6. The analytic launch collision avoidancemethod of claim 3, wherein generating the pair of times includescomputing corrections to the pair of times using a two variableminimization technique.
 7. The analytic launch collision avoidancemethod of claim 3, wherein generating the pair of times includescomputing corrections to the pair of times using Newton's method.
 8. Theanalytic launch collision avoidance method of claim 3, furthercomprising the step of: generating a starting time pair, as initialinputs for the process of refining the pair of times, depending uponchanges in sign of a distance function along an orbit.
 9. The analyticlaunch collision avoidance method of claim 8, wherein generating astarting time pair includes utilizing a root-finding algorithm to refinea candidate starting time pair and determine the time at which thesatellite and the launch vehicle come closest for a fixed discretelaunch time.
 10. The analytic launch collision avoidance method of claim9, wherein the root-finding algorithm is Brent's method.
 11. Theanalytic launch collision avoidance method of claim 3, wherein providinga launch collision avoidance report includes comparing a minimumdistance associated with the pair of times to a distance threshold todetermine whether a candidate launch time should be rejected.
 12. Theanalytic launch collision avoidance method of claim 3, wherein providinga launch collision avoidance report includes using the pair of times todetermine a collision probability.
 13. The analytic launch collisionavoidance method of claim 1, wherein evaluating the launch times and thecandidate satellite crossing times includes comparing a launch vehicletrajectory to the perigee and apogee of satellites to eliminatesatellites from further analysis.
 14. The analytic launch collisionavoidance method of claim 1, wherein evaluating the launch times and thecandidate satellite crossing times includes applying an altitude filterto an orbit associated with a candidate satellite crossing time.
 15. Theanalytic launch collision avoidance method of claim 1, whereinevaluating the launch times and the candidate satellite crossing timesincludes determining a satellite time window during which a satellitepasses through a launch wedge.
 16. The analytic launch collisionavoidance method of claim 15, wherein the launch wedge is formed byapproximating a plane followed by the trajectory of the launch vehicle.17. The analytic launch collision avoidance method of claim 1, whereinevaluating the launch times and the candidate satellite crossing timesincludes determining a launch vulnerability band across a launch wedgefrom an intersection of a satellite orbit with the launch wedge.
 18. Ananalytic launch collision avoidance method, comprising the steps of:evaluating launch times for a launch vehicle and candidate satellitecrossing times for a plurality of satellites using completely analyticpartial derivatives to determine unsafe launch times; and initiating orcanceling a launch event depending upon the unsafe launch times.
 19. Theanalytic launch collision avoidance method of claim 18, whereinevaluating the launch times and the candidate satellite crossing timesincludes generating a pair of times for which the launch vehicle and asatellite come closest together.
 20. The analytic launch collisionavoidance method of claim 19, wherein the pair of times is a launch timefor the launch vehicle and a real world time when the satellite and thelaunch vehicle come closest together.
 21. The analytic launch collisionavoidance method of claim 19, wherein generating the pair of timesincludes computing corrections to the pair of times using an iterativetechnique.
 22. The analytic launch collision avoidance method of claim19, wherein generating the pair of times includes computing correctionsto the pair of times using a two variable minimization technique. 23.The analytic launch collision avoidance method of claim 19, whereingenerating the pair of times includes computing corrections to the pairof times using Newton's method.
 24. The analytic launch collisionavoidance method of claim 19, further comprising the step of: generatinga starting time pair, as initial inputs for the process of generatingthe pair of times, depending upon changes in sign of a distance functionalong an orbit.
 25. The analytic launch collision avoidance method ofclaim 24, wherein generating a starting time pair includes utilizing aroot-finding algorithm to refine a candidate starting time pair anddetermine the time at which the satellite and the launch vehicle comeclosest for a fixed discrete launch time.
 26. The analytic launchcollision avoidance method of claim 25, wherein the root-findingalgorithm is Brent's method.
 27. The analytic launch collision avoidancemethod of claim 19, wherein evaluating the launch times and thecandidate satellite crossing times includes comparing a minimum distanceassociated with the pair of times to a distance threshold to determinewhether a candidate launch time should be rejected.
 28. The analyticlaunch collision avoidance method of claim 19, wherein evaluating thelaunch times and the candidate satellite crossing times includes usingthe pair of times to determine a collision probability.
 29. The analyticlaunch collision avoidance method of claim 18, wherein evaluating thelaunch times and the candidate satellite crossing times includescomparing a launch vehicle trajectory to the perigee and apogee ofsatellites to eliminate satellites from further analysis.
 30. Theanalytic launch collision avoidance method of claim 18, whereinevaluating the launch times and the candidate satellite crossing timesincludes applying an altitude filter to an orbit associated with acandidate satellite crossing time.
 31. The analytic launch collisionavoidance method of claim 18, wherein evaluating the launch times andthe candidate satellite crossing times includes determining a satellitetime window during which a satellite passes through a launch wedge. 32.The analytic launch collision avoidance method of claim 31, wherein thelaunch wedge is formed by approximating a plane followed by thetrajectory of the launch vehicle.
 33. The analytic launch collisionavoidance method of claim 18, wherein evaluating the launch times andthe candidate satellite crossing times includes determining a launchvulnerability band across a launch wedge from an intersection of asatellite orbit with the launch wedge.